Abstract

Near-forward scattering of an optically trapped 5-μm-radius polystyrene latex sphere by the trapping beam was examined both theoretically and experimentally. Since the trapping beam is tightly focused, the beam fields superpose and interfere with the scattered fields in the forward hemisphere. The observed light intensity consists of a series of concentric bright and dark fringes centered about the forward-scattering direction. Both the number of fringes and their contrast depend on the position of the trapping beam focal waist with respect to the sphere. The fringes are caused by diffraction that is due to the truncation of the tail of the trapping beam as the beam is transmitted through the sphere.

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  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett , 11, 288-290 (1986).
    [CrossRef] [PubMed]
  2. R. Gussgard, T. Lindmo, and I. Brevic, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am. B 9, 1922-1930 (1992).
    [CrossRef]
  3. A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
    [CrossRef] [PubMed]
  4. Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
    [CrossRef]
  5. A. Rohrbach and E. H. K. Stelzer, "Optical trapping of dielectric particles in arbitrary fields," J. Opt. Soc. Am. A 18, 839-853 (2001).
    [CrossRef]
  6. A. Rohrbach and E. H. K. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations," Appl. Opt. 41, 2494-2507 (2002).
    [CrossRef] [PubMed]
  7. In Ref. 13 it was incorrectly stated that Refs. 5 and 6 calculated the trapping force by use of Rayleigh scattering.
  8. J. S. Kim and S. S. Lee, "Scattering of laser beams and the optical potential well for a homogeneous sphere," J. Opt. Soc. Am. 73, 303-312 (1983).
    [CrossRef]
  9. G. Gouesbet, B. Maheu, and G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
    [CrossRef]
  10. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
    [CrossRef]
  11. K. F. Ren, G. Grehan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt. 35, 2702-2710 (1996).
    [CrossRef] [PubMed]
  12. K. F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
    [CrossRef]
  13. J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. 1. Localized model description of an on-axis tighty focused laser beam with spherical aberration," Appl. Opt. 43, 2532-2544 (2004).
    [CrossRef] [PubMed]
  14. J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. 2. On-axis trapping force," Appl. Opt. 43, 2545-2554 (2004).
    [CrossRef] [PubMed]
  15. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
    [CrossRef]
  16. P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
    [CrossRef]
  17. P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: An integral representation: Errata," J. Opt. Soc. Am. A 12, 1605 (1995).
  18. J.-P. Chevaillier, J. Fabre, G. Grehan, and G. Gouesbet, "Comparison of diffraction theory and generalized Lorenz-Mie theory for a sphere located on the axis of a laser beam," Appl. Opt. 29, 1293-1298 (1990).
    [CrossRef] [PubMed]
  19. F. Gilloteau, G. Grehan, and G. Gouesbet, "Optical levitation experiments to assess the validity of the generalized Lorenz-Mie theory," Appl. Opt. 31, 2942-2951 (1992).
    [CrossRef]
  20. J. A. Lock and J. T. Hodges, "Far-field scattering of an axisymmetric laser beam of arbitrary profile by an on-axis spherical particle," Appl. Opt. 35, 4283-4290 (1996).
    [CrossRef] [PubMed]
  21. J. A. Lock and J. T. Hodges, "Far-field scattering of a non-Gaussian off-axis axisymmetric laser beam by a spherical particle," Appl. Opt. 35, 6605-6616 (1996).
    [CrossRef] [PubMed]
  22. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 604, Eqs. (11.85) and (11.86).
  23. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 123.
  24. H. M. Nussenzveig, "High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission," J. Math. Phys. 10, 82-124 (1969).
    [CrossRef]
  25. E. A. Hovenac and J. A. Lock, "Assessing the contributions of surface waves and complex rays to the far-field Mie scattering by use of the Debye series," J. Opt. Soc. Am. A 9, 781-795 (1992).
    [CrossRef]
  26. L. Brillouin, "The scattering cross section of spheres for electromagnetic waves," J. Appl. Phys. 20, 1110-1125 (1949).
    [CrossRef]
  27. B. Maheu, G. Grehan, and G. Gouesbet, "Ray localization in Gaussian beams," Opt. Commun. 70, 259-262 (1989).
    [CrossRef]
  28. J. A. Lock, "Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle," J. Opt. Soc. Am. A 10, 693-706 (1993).
    [CrossRef]
  29. M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 487, Eq. (11.4.42).
  30. H. M. Nussenzveig and W. J. Wiscombe, "Forward optical glory," Opt. Lett. 5, 455-457 (1980).
    [CrossRef] [PubMed]
  31. D. S. Langley and M. J. Morrell, "Rainbow-enhanced forward and backward glory scattering," Appl. Opt. 30, 3459-3467 (1991).
    [CrossRef] [PubMed]
  32. J. A. Lock and T. A. McCollum, "Further thoughts on Newton's zero-order rainbow," Am. J. Phys. 62, 1082-1089 (1994).
    [CrossRef]
  33. R. C. Weast (ed.), Handbook of Chemistry and Physics, 50th ed. (CRC, 1969), p. F36.
  34. J. Happel and H. Bremmer, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, 1965), p. 330, Eqs. (7.4.37)-(7.4.39) and p. 331, Table 7.4.1.

2004

2002

2001

1996

1995

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: An integral representation: Errata," J. Opt. Soc. Am. A 12, 1605 (1995).

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
[CrossRef]

1994

K. F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

J. A. Lock and T. A. McCollum, "Further thoughts on Newton's zero-order rainbow," Am. J. Phys. 62, 1082-1089 (1994).
[CrossRef]

1993

1992

1991

1990

1989

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

B. Maheu, G. Grehan, and G. Gouesbet, "Ray localization in Gaussian beams," Opt. Commun. 70, 259-262 (1989).
[CrossRef]

1988

1986

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett , 11, 288-290 (1986).
[CrossRef] [PubMed]

1985

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 604, Eqs. (11.85) and (11.86).

1983

1981

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 123.

1980

1969

R. C. Weast (ed.), Handbook of Chemistry and Physics, 50th ed. (CRC, 1969), p. F36.

H. M. Nussenzveig, "High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission," J. Math. Phys. 10, 82-124 (1969).
[CrossRef]

1965

J. Happel and H. Bremmer, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, 1965), p. 330, Eqs. (7.4.37)-(7.4.39) and p. 331, Table 7.4.1.

1964

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 487, Eq. (11.4.42).

1959

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

1949

L. Brillouin, "The scattering cross section of spheres for electromagnetic waves," J. Appl. Phys. 20, 1110-1125 (1949).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 487, Eq. (11.4.42).

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 604, Eqs. (11.85) and (11.86).

Asakura, T.

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Ashkin, A.

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett , 11, 288-290 (1986).
[CrossRef] [PubMed]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Bjorkholm, J. E.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett , 11, 288-290 (1986).
[CrossRef] [PubMed]

Booker, G. R.

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
[CrossRef]

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: An integral representation: Errata," J. Opt. Soc. Am. A 12, 1605 (1995).

Bremmer, H.

J. Happel and H. Bremmer, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, 1965), p. 330, Eqs. (7.4.37)-(7.4.39) and p. 331, Table 7.4.1.

Brevic, I.

Brillouin, L.

L. Brillouin, "The scattering cross section of spheres for electromagnetic waves," J. Appl. Phys. 20, 1110-1125 (1949).
[CrossRef]

Chevaillier, J.-P.

Chu, S.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett , 11, 288-290 (1986).
[CrossRef] [PubMed]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett , 11, 288-290 (1986).
[CrossRef] [PubMed]

Fabre, J.

Gilloteau, F.

Gouesbet, G.

Grehan, G.

Gussgard, R.

Happel, J.

J. Happel and H. Bremmer, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, 1965), p. 330, Eqs. (7.4.37)-(7.4.39) and p. 331, Table 7.4.1.

Harada, Y.

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Hodges, J. T.

Hovenac, E. A.

Kim, J. S.

Laczik, Z.

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: An integral representation: Errata," J. Opt. Soc. Am. A 12, 1605 (1995).

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
[CrossRef]

Langley, D. S.

Lee, S. S.

Lindmo, T.

Lock, J. A.

Maheu, B.

McCollum, T. A.

J. A. Lock and T. A. McCollum, "Further thoughts on Newton's zero-order rainbow," Am. J. Phys. 62, 1082-1089 (1994).
[CrossRef]

Morrell, M. J.

Nussenzveig, H. M.

H. M. Nussenzveig and W. J. Wiscombe, "Forward optical glory," Opt. Lett. 5, 455-457 (1980).
[CrossRef] [PubMed]

H. M. Nussenzveig, "High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission," J. Math. Phys. 10, 82-124 (1969).
[CrossRef]

Ren, K. F.

K. F. Ren, G. Grehan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Rohrbach, A.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 487, Eq. (11.4.42).

Stelzer, E. H. K.

Torok, P.

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
[CrossRef]

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: An integral representation: Errata," J. Opt. Soc. Am. A 12, 1605 (1995).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 123.

Varga, P.

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
[CrossRef]

P. Torok, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: An integral representation: Errata," J. Opt. Soc. Am. A 12, 1605 (1995).

Weast, R. C.

R. C. Weast (ed.), Handbook of Chemistry and Physics, 50th ed. (CRC, 1969), p. F36.

Wiscombe, W. J.

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Am. J. Phys.

J. A. Lock and T. A. McCollum, "Further thoughts on Newton's zero-order rainbow," Am. J. Phys. 62, 1082-1089 (1994).
[CrossRef]

Appl. Opt.

D. S. Langley and M. J. Morrell, "Rainbow-enhanced forward and backward glory scattering," Appl. Opt. 30, 3459-3467 (1991).
[CrossRef] [PubMed]

F. Gilloteau, G. Grehan, and G. Gouesbet, "Optical levitation experiments to assess the validity of the generalized Lorenz-Mie theory," Appl. Opt. 31, 2942-2951 (1992).
[CrossRef]

K. F. Ren, G. Grehan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

J. A. Lock and J. T. Hodges, "Far-field scattering of an axisymmetric laser beam of arbitrary profile by an on-axis spherical particle," Appl. Opt. 35, 4283-4290 (1996).
[CrossRef] [PubMed]

J. A. Lock and J. T. Hodges, "Far-field scattering of a non-Gaussian off-axis axisymmetric laser beam by a spherical particle," Appl. Opt. 35, 6605-6616 (1996).
[CrossRef] [PubMed]

J.-P. Chevaillier, J. Fabre, G. Grehan, and G. Gouesbet, "Comparison of diffraction theory and generalized Lorenz-Mie theory for a sphere located on the axis of a laser beam," Appl. Opt. 29, 1293-1298 (1990).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations," Appl. Opt. 41, 2494-2507 (2002).
[CrossRef] [PubMed]

J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. 1. Localized model description of an on-axis tighty focused laser beam with spherical aberration," Appl. Opt. 43, 2532-2544 (2004).
[CrossRef] [PubMed]

J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. 2. On-axis trapping force," Appl. Opt. 43, 2545-2554 (2004).
[CrossRef] [PubMed]

Biophys. J.

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

J. Appl. Phys.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

L. Brillouin, "The scattering cross section of spheres for electromagnetic waves," J. Appl. Phys. 20, 1110-1125 (1949).
[CrossRef]

J. Math. Phys.

H. M. Nussenzveig, "High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission," J. Math. Phys. 10, 82-124 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

B. Maheu, G. Grehan, and G. Gouesbet, "Ray localization in Gaussian beams," Opt. Commun. 70, 259-262 (1989).
[CrossRef]

K. F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Opt. Lett

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett , 11, 288-290 (1986).
[CrossRef] [PubMed]

Opt. Lett.

Proc. R. Soc. London A

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Other

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 604, Eqs. (11.85) and (11.86).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 123.

In Ref. 13 it was incorrectly stated that Refs. 5 and 6 calculated the trapping force by use of Rayleigh scattering.

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 487, Eq. (11.4.42).

R. C. Weast (ed.), Handbook of Chemistry and Physics, 50th ed. (CRC, 1969), p. F36.

J. Happel and H. Bremmer, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Prentice-Hall, 1965), p. 330, Eqs. (7.4.37)-(7.4.39) and p. 331, Table 7.4.1.

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Figures (8)

Fig. 1
Fig. 1

Far-zone beam intensity (solid curve) as a function of the scattering angle in water θ for the freely propagating Gaussian beam generated from the shape coefficients of Eqs. (17) and (18) with n = 1.33, λ = 0.532 μm, w = 0.172 μm, wa = 0.205 μm, z 0 = −1.60 μm, and l max = 97. The dashed curve is the far-zone paraxial approximation of Eq. (12) with the intended width w.

Fig. 2
Fig. 2

Far-zone beam intensity (solid curve) as a function of the scattering angle in water θ2 for the AFA beam generated from the shape coefficients of Eqs. (19)–(22) with W∕A = 5.0, n = 1.50, n 2 = 1.33, λ = 0.532 μm, NA = 1.25, z 0 = 0.13 μm, d = −4.95 μm, and l max = 1200. The beam cuts off at θ2 = 53.8°. The dashed curve is the far-zone paraxial approximation of Eq. (23), and the open circles are the reconstructed intensity when l max = 97.

Fig. 3
Fig. 3

Far-zone beam-plus-scattered intensity (solid curve) as a function of the scattering angle in water θ for the freely propagating Gaussian beam of Fig. 1 incident upon a PSL sphere with a = 4.987 μm and N = 1.59 for (a) z 0 = −1.60 μm, (b) z 0 = −4.32 μm, and (c) z 0 = −7.72 μm. The dashed curve is the intensity for the transmitted term of the Debye series expansion of the scattered light.

Fig. 4
Fig. 4

Far-zone beam-plus-scattered intensity (solid curve) for θ = 0° as a function of the beam focal point location z 0 for the freely propagating Gaussian beam of Fig. 1 incident upon the PSL sphere of Fig. 3. The dashed curve is the θ = 0° intensity for the transmitted term of the Debye series expansion of the scattered light.

Fig. 5
Fig. 5

Far-zone beam-plus-scattered intensity (solid curve) as a function of the scattering angle in water θ2 for the AFA beam of Fig. 2 (except with W∕A = 1.5 and new values of d based on the value of z 0) incident upon the PSL sphere of Fig. 3 for (a) z 0 = 0.08 μm, (b) z 0 = −2.52 μm, and (c) z 0 = −6.30 μm. The dashed curve is the intensity for the transmitted term of the Debye series expansion of the scattered light.

Fig. 6
Fig. 6

Far-zone beam-plus-scattered intensity (solid curve) for θ2 = 0° as a function of the beam focal point location z 0 for the AFA beam of Fig. 5 incident upon the PSL sphere of Fig. 3. The dashed curve is the θ2 = 0° intensity for the transmitted term of the Debye series expansion of the scattered light.

Fig. 7
Fig. 7

Measured trapping length of the laser beam as a function of the distance of the PSL sphere's surface from the bottom of the glass coverslip, based on a 100% retrapping rate.

Fig. 8
Fig. 8

Near-forward beam-plus-scattered intensity at various times as the PSL sphere is pulled back up to its stable trapping position: (a) 0.000 s, (b) 0.915 s, (c) 1.213 s, (d) 1.426 s, (e) 1.612 s, (f) 1.801 s, (g) 2.305 s, and (h) 2.615 s.

Equations (215)

Equations on this page are rendered with MathJax. Learn more.

5 -μm
k = 2 π / λ
exp ( i ω t )
z = z 0
E 0
E beam ( r , θ , ϕ ) = E 0 l = 1 i l + 1 ( 2 l + 1 ) g l [ j l ( n k r ) / ( n k r ) ] ×  π l ( θ ) sin ( θ ) cos ( ϕ ) u r + E 0 l = 1 { i l ( 2 l + 1 ) / [ l ( l + 1 ) ] } ×  [ h l j l ( n k r ) π l ( θ ) i g l L l ( n k r ) τ l ( θ ) ] ×  cos ( ϕ ) u θ E 0 l = 1 { i l ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ h l j l ( n k r ) τ l ( θ ) i g l L l ( n k r ) π l ( θ ) ] sin ( ϕ ) u ϕ ,
g l   and   h l
j l ( n k r )
L l ( n k r ) j l ( n k r ) / ( n k r ) j l ( n k r ) ,
π l ( θ ) = P l     1 [ cos ( θ ) ] / sin ( θ ) ,
τ l ( θ ) = ( d / d θ ) P l     1 [ cos ( θ ) ] .
g l = h l
r
E beam ( r , θ , ϕ ) = i E 0 [ exp ( i n k r ) / ( n k r ) ] × [ S 2 , beam ( θ ) cos ( ϕ ) u θ + S 1 , beam ( θ ) sin ( ϕ ) u ϕ ] + O ( 1 / r 2 ) ,
S 1 , beam ( θ ) = ( - ½ ) l = 1 { ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ g l π l ( θ ) + h l τ l ( θ ) ] ,
S 2 , beam ( θ ) = ( - ½ ) l = 1 { ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ g l τ l ( θ ) + h l π l ( θ ) ] .
E scatt ( r , θ , ϕ ) = i E 0 [ exp ( i n k r ) / ( n k r ) ] × [ S 2 , scatt ( θ ) cos ( ϕ ) u θ + S 1,scatt ( θ ) sin ( ϕ ) u ϕ ] + O ( l / r 2 ) ,
S 1 , scatt ( θ ) = l = 1 { ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ a l g l π l ( θ ) + b l h l τ l ( θ ) ] ,
S 2 , scatt ( θ ) = l = 1 { ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ a l g l τ l ( θ ) + b l h l π l ( θ ) ] ,
a l
b l
S 1 , total ( θ ) = S 1 , beam ( θ ) + S 1 , scatt ( θ ) ,
S 2 , total ( θ ) = S 2 , beam ( θ ) + S 2 , scatt ( θ ) .
2 π n a / λ 1
a l , b l = ( ½ ) [ 1 R l     external p = 1 T l     in ( R l     internal ) p 1 T l     out ] ,
( R l external / 2 )
( T l in T l out / 2 )
p 1
[ ( R l     internal ) p 1 ]
s 1 / ( n k w ) .
s 1
g l
h l
ρ
z = 0
l + ½
z = 0
E beam ( ρ , 0 ) = E 0     exp ( ρ 2 / w 2 ) u x ,
E diffracted ( r , θ , ϕ ) = ( i E 0 / 2 s 2 ) [ exp ( i n k r ) / ( n k r ) ] × exp ( θ 2 / 4 s 2 ) [ cos ( ϕ ) u θ + sin ( ϕ ) u ϕ ]
g l = h l = exp [ s 2 ( l + ½ ) 2 ] .
z = 0
E beam ( ρ , 0 )  =  2 E 0 J 1 ( n k ρ α ) / ( n k ρ α ) u x ,
E diffracted ( r , θ , ϕ ) = [ 2 i E 0 / α 2 ] [ exp ( i n k r ) / ( n k r ) ] × [ cos ( ϕ ) u θ + sin ( ϕ ) u ϕ ]   for   θ < α
= 0  for  θ > α ,
g l = h l = 2 J 1 [ ( l + ½ ) α ] / [ ( l + ½ ) α ] .
z = z 0
z = 0
z = z 0
z = 0
s 1
s 1
z 0 0
θ α
s 1
w a
g l
h l
z = z 0
g l = h l = D exp ( i n k z 0 ) exp [ D s 2 ( l + ½ ) 2 ] ,
D = ( 1 2 i s z 0 / w ) 1 .
n = 1.33
λ = 0.532   μm
w = 0.172   μm
w a = 0.205   μm
NA=1 .25
z 0
z 0 = 1.60   μm
l max = 97
0 ° θ 60 °
w a
n 1
n 2
z 0
d < z 0
g l = ( i n 1 k F / 2 ) 0 α sin ( θ 1 ) d θ 1 [ cos ( θ 1 ) ] ½ × exp { i [ n 2 k d cos ( θ 2 ) + n 1 k ( z 0 d ) cos ( θ 1 ) ] } × exp [ ( A / W ) 2 tan 2 ( θ 1 ) / tan 2 ( α ) ] × { [ t TE + t TM cos ( θ 2 ) ] J 0 [ ( l + ½ ) sin ( θ 2 ) ] + [ t TE t TM cos ( θ 2 ) ] J 2 [ ( l + ½ ) sin ( θ 2 ) ] } ,
h l = ( i n 1 k F / 2 ) 0 α sin ( θ 1 ) d θ 1 [ cos ( θ 1 ) ] ½ × exp { i [ n 2 k d cos ( θ 2 ) + n 1 k ( z 0 d ) cos ( θ 1 ) ] } × exp [ ( A / W ) 2 tan 2 ( θ 1 ) / tan 2 ( α ) ] × { [ t TM + t TE cos ( θ 2 ) ] J 0 [ ( l + ½ ) sin ( θ 2 ) ] + [ t TM t TE cos ( θ 2 ) ] J 2 [ ( l + ½ ) sin ( θ 2 ) ] } ,
NA = n 1 sin ( α ) ,
θ 1
θ 2
n 1 sin ( θ 1 ) = n 2 sin ( θ 2 ) ,
t TE = 2 cos ( θ 1 ) / [ cos ( θ 1 ) + ( n 2 / n 1 ) cos ( θ 2 ) ] ,
t TM = 2 cos ( θ 1 ) / [ ( n 2 / n 1 ) cos ( θ 1 ) + cos ( θ 2 ) ] .
W / A = 5.0
n 1 = 1.50
n 2 = 1.33
λ = 0.532   μm
NA=1 .25
α =56 .4 °
0 .202   μm
l max = 1200
z 0 = 0.13   μm
| S 1 , diffracted ( θ ) | 2 = ( n 2 / n 1 ) 4 × [ t TE + t TM cos ( θ 2 ) ] 2 / [ 4 cos ( θ 1 ) ] ,
l max = 97
α = 56.4 °
θ 2 = 69.9 °
θ 2 = 53.8 °
θ 2 = ( n 1 / n 2 ) sin ( α )
30 ° α 60 °
W / A 1.5
n = 1.33
λ = 0.532   μm
w = 0.172   μm
w a = 0.205   μm
z 0
a = 4.987   μm
N = 1.59
8.4   μm z 0 1.5   μm
z 0 5   μm
z 0 5   μm
Q max = 0.0313
z 0 = 4.8   μm
60   mW
1 .05   g / cm 3
z 0 = 1.5   μm
2   mW
z 0 = 4.8   μm
z 0 1.5   μm
6 .9   μm
n 1 = 1.50
n 2 = 1.33
W / A = 1.5
λ = 0.532   μm
NA = 1.25
d = z 0 5.08   μm
6 .30   μm z 0 0.08   μm
z 0 p
z 0 e
6.88   μm z 0 p 0.50   μm
7.57   μm z 0 e 1.19   μm
60 %
Q max = 0.0267
z 0 = 2.34   μm
z 0 0.08   μm
6.38   μm
a = 4.987   μm
40   μm d 5   μm
40 ° 50 °
a = 4.987   μm
z 0
z 0
θ 40 °
z 0 = 1.60   μm
w ( z ) = w [ 1 + 4 ( z z 0 ) 2 / ( n k w 2 ) 2 ] 1 / 2 ,
z 0 = 1.60   μm
z 0 = 4.32   μm
z 0 w / 2 s
g l = h l ( i w / 2 s z 0 ) exp ( i n k z 0 )
×  exp [ i s w ( l + ½ ) 2 / 2 z 0 ] ×  exp [ w 2 ( l + ½ ) 2 / 4 z 0     2 ] .
a = 4.987   μm
l max = 1 + ( 2 π n a / λ ) + 4.3 ( 2 π n a / λ ) 1 / 3 = 97.
z 0 = 1.6   μm
l max = 97   is   10 12
z 0 = 4.32 μm
l max
z 0
z 0
42.4 °
l / e 2
z 0 = 1.60   μm
θ = 38.3 °
z 0 = 4.32   μm
θ =29 .4 °
l / e 2
2 π n a / λ = 78
z = z 0
z = 0
z 0 = 4.32   μm
z 0 = 7.72   μm
z 0 15.3   μm
θ =9 .7 °
z 0 = 7.72   μm
θ = 0 °
z 0
( z 0 4   μm )
( z 0 5   μm )
z 0 = 5.85 , 7.45 ,
11.28   μm
z 0 = 5.33 , 6.41 ,   and   8.31   μm
Δ L
Δ L
0.5 λ
0.71 λ , 0.58 λ , 0.75 λ , 0.43 λ ,   and   0.76 λ
θ = 0 °
z 0 = 0.08   μm
42 °
l max = 97
a = 4.987   μm
z = 0
z 0 = 2.52   μm
z 0 = 6.30   μm
θ = 0 °
z 0
3.17   mm
a = 4.987 ± 0.030   μm
10 −3
123 ± 3   and   140 ± 4   μm
λ =0 .532   μm
25   mW
NA = 1.25 100 ×
20   ° C   to   34   ° C
2.7   and   3.8 μm / s
4   and   50   μm
100 %
Δ = 4   μm
23   μm
7  μm
100%
Δ > 30   μm
Δ =4   and   5   μm
Δ =4   μm
1 .6
Δ =10   μm
1 .1
20   s
z 0

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